Evaluate where is the circle with centre 0 and radius 2, traversed in the usual anti-clockwise sense.

And my answer...

The function is holomorphic on the complex plane except at z=1.

So, using Cauchys integral formula we get with .

Hence, .

This right?

Yes

If so, how come the answer doesnt depend on ? Does it not matter what the radius of the circle is?

It does depend on it.
Because the pole z=1 is within the circle, you get the result.
If it wasn't (say you considered as the circle of center 5 and radius 1), then the integral would have been 0.
Look here : Cauchy's integral formula - Wikipedia, the free encyclopedia (they say 'Then for every a in the interior of D')